5-dimensional geometries I: the general classification

Andrew Geng

June 9, 2016

Abstract

This paper is the first of a 3-part series that classifies the 5-dimensional Thurston geometries.The present paper (part 1 of 3) summarizes the general classification, giving the full list, anoutline of the method, and some illustrative examples. This includes phenomena that havenot appeared in lower dimensional geometries, such as an uncountable family of geometriesS̃L2 ×α S

3.

Contents

1 Introduction 1

2 Salient examples 42.1 New phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Examples that highlight tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Examples that clarify how the classification is organized . . . . . . . . . . . . . . . . 6

3 Overview of method 7

4 Related work 10

5 References 12

1 Introduction

By the classification of closed surfaces (see e.g. [Mat02, Thm. 5.11]), every closed surface is diffeo-morphic to a quotient of E2, S2, or H2 by a discrete group of isometries. It is a classical resultthat in dimension 2, these three are the only connected, simply-connected, complete Riemannianmanifolds with transitive isometry group (see e.g. [Thu97, Thm. 3.8.2]).

The quest for the 3-dimensional generalization that became Thurston’s Geometrization Conjec-ture led to a version of the following definition. (The equivalence to older definitions is outlined inPart II, [Gen16a, Prop. 2.5].)

Definition 1.1 (Geometries, following [Thu97, Defn. 3.8.1] and [Fil83, §1.1]).

(i) A geometry is a connected, simply-connected homogeneous space M = G/Gp where G is aconnected Lie group acting faithfully with compact point stabilizers Gp.

(ii) M is a model geometry if there is some lattice Γ ⊂ G that acts freely onM . Then the manifoldΓ\G/Gp is said to be modeled on M .

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(iii) M is maximal if it is not G-equivariantly diffeomorphic to any other geometry G′/G′p withG ( G′. Any such G′/G′p is said to subsume G/Gp.

Then a closed 3-manifold is a quotient of at most one maximal model geometry, which canbe determined from the fundamental group [Thu97, Thm. 4.7.8] or from the existence of certainbundle structures (usually Seifert bundles) and some topological data (usually two Euler numbers)[Sco83, Thm. 5.3]. Thurston classified the 3-dimensional maximal model geometries and found eight(see [Thu97, Thm. 3.8.4]).

In 4 dimensions, Filipkiewicz classified the maximal model geometries in [Fil83]. Though 4-manifolds without geometric decompositions [Hil02, §13.3 #3] indicate there is less hope for astraightforward generalization of geometrization, Filipkiewicz’s classification highlights a few inter-esting firsts. The list comprises 18 geometries and—for the first time—a countably infinite family,named Sol4m,n. (See e.g. [Hil02, §7.1] or [Wal86, §1, Table 1] for the names currently in use.) One ofthe eighteen is F4 = R2 o SL(2,R)/SO(2), the first geometry to admit finite-volume quotients butno compact quotients.

The direction to take should now seem straightforward. One seeks a classification of maxi-mal model geometries in all dimensions; but a handful of obstacles stand in the way of such aclassification:

1. Existing classifcations, including now the present paper, rely on tools that may become un-usable with increasing dimension. For example, the case of discrete point stabilizers ([Thu97,Thm. 3.8.4(c)] in dimension 3, [Fil83, Ch. 6] in dimension 4, and [Gen16a, Thm. 1.1(ii)] in PartII) relies on a classification of solvable Lie algebras over R, which is incomplete in dimensions7 and up. (See e.g. [ŠW12, Introduction] for a summary of known progress, and [BFNT13]for a wider survey.)

2. The aforementioned aspects of the 4-dimensional classification suggest that new phenomenamay continue to appear for a few more dimensions. A workable approach to a general classi-fication may not be evident without knowledge of such features.

An optimistic interpretation of these obstacles is that the 5-dimensional case is both tractable andpotentially illustrative. Having carried out the classification, the new phenomena are summarizedin Section 2; the main result is the following list.

Theorem 1.2 (Classification of 5-dimensional geometries). The maximal model geometriesof dimension 5 are:

1. The geometries with constant curvature:

E5 = R5 o SO(5)/ SO(5) S5 = SO(6)/ SO(5) H5 = SO(5, 1)/ SO(5);

2. The other irreducible Riemannian symmetric spaces SL(3,R)/ SO(3) and SU(3)/ SO(3);

3. The unit tangent bundles or universal covers of circle bundles:

T 1(H3) = PSL(2,C)/ SO(2) T 1(E1,2) = R3 o SO(1, 2)0/ SO(2) ˜U(2, 1)/U(2);

2

4. The associated bundles (see e.g. [Sha00, §1.3 Vector Bundles] for the notation):

Heis3 ×R S3 = (Heis3 o S̃O(2))× S3/{(0, 0, s), γ(t), eπis}s,t∈R

Heis3 ×R S̃L2 = (Heis3 o S̃O(2))× S̃L2/{(0, 0, s), γ(t), γ(s)}s,t∈RS̃L2 ×α S3 = S̃L2 × S3 × R/{γ(s), eπit, αs+ t}s,t∈R, 0 < α <∞

S̃L2 ×α S̃L2 = S̃L2 × S̃L2 × R/{γ(s), γ(t), αs+ t}s,t∈R, 0 < α ≤ 1

L(a; 1)×S1 L(b; 1) = S3 × S3 × R/{eπis, eπit, as+ bt}s,t∈R, 0 < a ≤ b coprime in Z,

where the Heisenberg group Heis3 is R3 with the multiplication law

(x, y, z)(x′, y′, z′) = (x+ x′, y + y′, z + z′ + xy′ − x′y),

on which SO(2) acts through the action of SL(2,R) on the x, y plane, and t 7→ eπit ∈ S3 andγ : R→ S̃O(2) ⊂ S̃L2 are 1-parameter subgroups sending Z to the center;

5. The three principal R-bundles with non-flat connections over the F4 geometry, distinguishedfrom each other by their curvatures:

R2 o S̃L2∼= (R2 o S̃L2) o SO(2)/SO(2)

F5a = Heis3 o S̃L2/{(0, 0, at), γ(t)}t∈R, a = 0 or 1;

6. The six simply-connected indecomposable nilpotent Lie groups, named by their Lie algebrasas in [PSWZ76, Table II], in which the point stabilizer of the identity element is a maximalcompact group of automorphisms (specified in Table 3.2):

A5,1 = R4 oRx2, x2

A5,2 = R4 oRx4

A5,3 = (R×Heis3) oRx3→x2→y

A5,4 = Heis5 A5,5 = Nil4 oR3→1

A5,6 = Nil4 oR4→3→1

;

7. The simply-connected indecomposable non-nilpotent solvable Lie groups, specified the sameway:

Aa,b,−1−a−b5,7 = R4 oR4 distinct real roots

A1,−1−a,−1+a5,7 = R4 oR

2 complex, 2 distinct real

A1,−1,−15,7 = R4 oR

x−1, x−1, x+1, x+1A−15,8 = R4 oR

x2, x−1, x+1

A−1,−15,9 = R4 oR(x−1)2, x+1, x+1

A−15,15 = R4 oR(x−1)2, (x+1)2

A05,20 = (R×Heis3) oR

Lorentz, y→x1A−1,−15,33 = R3 o {xyz = 1}0;

8. and all twenty-nine products of lower-dimensional geometries involving no more than oneEuclidean factor, named as in [Wal86, Table 1].

(a) 4-by-1:

S4 × E H4 × E CP2 × E CH2 × E F4 × ENil4 × E Sol40 × E Sol41 × E Sol4m,n × E

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(b) 3-by-2:

E3 × S2 E3 ×H2

S3 × E2 S3 × S2 S3 ×H2

H3 × E2 H3 × S2 H3 ×H2

Heis3 × E2 Heis3 × S2 Heis3 ×H2

Sol3 × E2 Sol3 × S2 Sol3 ×H2

S̃L2 × E2 S̃L2 × S2 S̃L2 ×H2

(c) 2-by-2-by-1:

S2 × S2 × E S2 ×H2 × E H2 ×H2 × E

More explicit instructions for constructing these geometries—such as the solvable Lie groupsand their automorphism groups—are delegated to where they occur in the classification in Parts IIand III [Gen16a,Gen16b].

Roadmap. The present paper (Part I) summarizes the classification; Section 2 picks out illustra-tive examples, Section 3 outlines the strategy, and Section 4 briefly surveys related classifications.Part II [Gen16a] classifies the point stabilizer subgroups Gp and classifies the geometries where Gpacts irreducibly or trivially on tangent spaces. Part III [Gen16a] classifies the remaining geome-tries after showing that they all admit invariant fiber bundle structures (hence the name “fiberinggeometries”).

Acknowledgments. I have the pleasure to thank my advisor Benson Farb for years of helpful dis-cussions, patient advice, and extensive comments on drafts of all three papers in this series. Thanksalso to my second advisor Danny Calegari, especially for his keen sense of where the interestingunanswered questions lay; to Jonathan Hillman, Christoforos Neofytidis, Daniel Studenmund, IlyaGrigoriev, and Nick Salter for a number of other enlightening conversations; to Ilka Agricola forpointing out related work on SO(3)5-structures; and to Greg Friedman, Christoforos Neofytidis, andKenneth Knox for opportunities to speak about an early version of these results. Finally, I wouldlike to thank the University of Chicago for support during this work.

2 Salient examples

2.1 New phenomena

Much of our interest in this classification is in the search for phenomena that occur for the first timein dimension 5, in hopes of finding a pattern that continues in higher dimensions. See also Section3 for a discussion of new tools.

An uncountable family of geometries. The associated bundles S̃L2 ×α S3 (0 < α <∞) forman uncountable family of maximal model geometries. (Taking S̃L2 ×α S3 as a circle bundle overS2 × H2 with an invariant connnection, the parameter α is a ratio of curvatures in the S2 and H2

directions.) This and S̃L2 ×α S̃L2 are the first occurrences of uncountable families. Since everylattice in a Lie group is finitely presented [OV00, Thm. I.1.3.1], π1 of the quotient manifolds willnot determine the geometries. Details are in Part III, [Gen16b, Prop. 6.35].

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An infinite family without compact quotients. In fact, S̃L2×α S3 admits compact quotientsif and only if α is rational [Gen16b, Prop. 6.36]. (Recall that beginning with H2, geometries canhave noncompact quotients of finite volume; and beginning with F4 = R2 o SL(2,R)/ SO(2), modelgeometries might have no compact quotients.)

Non-unique maximality. The geometry T 1S3 = SO(4)/SO(2) is a non-maximal form of S3×S2

and L(1; 1)×S1 L(1; 1)—both of which are maximal [Gen16b, Rmk. 6.40]. This contrasts with thepositive results for unique maximality listed in the discussion after [Fil83, Prop. 1.1.2].

Inequivalent compact geometries with the same diffeomorphism type. Using Barden’sdiffeomorphism classification [Bar65] of simply-connected 5-manifolds by second homology and sec-ond Stiefel-Whitney class, one can prove that the associated bundles of lens spaces L(a; 1)×S1L(b; 1)are all diffeomorphic to S3 × S2 [Ott09, Cor. 3.3.2].

More broadly one can attempt to give the classification up to diffeomorphism, following previousresults such as [Mos50, Cor. p. 624], [Gor77], [Ish55], and [Ott09, Thm. 1.0.3]. Most of the geometriesare products of Rk and some spheres; the two exceptions are CP2 × E and the rational homologysphere SU(3)/ SO(3), named X−1 in Barden’s classification [BG02, Introduction].

Note that while the correct diffeomorphism type may be obvious enough to guess, it is notas obviously correct. The problem is proving that the space is a direct product of Rk and theproduct of spheres onto which it deformation retracts—such a claim is false for any nontrivialvector bundle (such as TS2) and for a homogeneous example by Samelson discussed in [Mos55, §5Example 4]. Instead one has to use either an explicit description of the diffeomorphism type from[Mos62, Thm. A] or the fact that sufficiently nice bundles over contractible spaces are trivial [Hus94,Cor. 10.3].

Isotropy irreducible spaces. The geometries SL(3,R)/ SO(3) and SU(3)/ SO(3) have point sta-bilizers SO(3) acting irreducibly on the (5-dimensional) tangent spaces. This is the first occurrenceof such an action being irreducible and not the standard representation for a group of the sameisomorphism type. These two geometries are still symmetric spaces, but in higher dimensions thereexist homogeneous spaces with irreducibly-acting point stabilizers that are not symmetric spaces(See e.g. [HZ96, Introduction]).

2.2 Examples that highlight tools

The classification of geometries requires an increasingly wide range of tools as the dimension in-creases. These are a handful of examples where either unexpected tools appeared, or familiar toolsexhibit behavior that is not completely obvious at first glance.

Model geometries via Galois theory and Dirichlet’s unit theorem. Some Galois theory isneeded to answer questions of lattice existence, such as to prove that Ro Conf+ E3/ SO(3) (wherethe action of Conf+ E3 on R is chosen to make the semidirect product unimodular) is not a modelgeometry [Gen16b, Prop. 5.1(iv)].

Dirichlet’s unit theorem makes an appearance when we construct a lattice in R3 o {xyz = 1}0by taking a finite index subgroup of OK o O×K where K is a totally real cubic field extension ofQ [Gen16a, Prop. 5.16].

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Point stabilizers not realized. The classification of geometries starts by classifying subgroupsof SO(5) in order to classify point stabilizers—but not every subgroup is realized by a maximalmodel geometry. For example, SO(3) in its standard representation is one such subgroup, thoughthe non-model geometry R o Conf+ E3/ SO(3) mentioned above suggests this can be thought ofas a near miss. The non-occurrence of SU(2) is another example, a feature shared by the 4-dimensional classification of geometries. Other subgroups—namely SO(4) and SO(3)× SO(2)—arepoint stabilizers only of product geometries. A listing of (non-product) geometries by point stabilizeris given below in Table 3.2 after Figure 3.1 names the subgroups.

Geometries in higher dimensions with reducible isotropy and no fibering. When pointstabilizers act reducibly on tangent spaces, our strategy breaks down the problem by showing theexistence of an invariant fiber bundle structure. That this is possible is is a convenient accident oflow dimensions; higher dimensions introduce isotropy-reducible geometries that admit no fibering.

For example, in dimension 18, there is Sp(3)/ Sp(1), where the embedding Sp(1) ↪→ Sp(3) isgiven by the irreducible representation of Sp(1) ∼= SU(2) on C6. This has two isotropy summands butadmits no nontrivial fibering since Sp(1) is maximal (so no larger group can be a point stabilizer ofthe base space) [DK08, Example V.10]. A strategy that continues to break the problem down usinginvariant fiber bundle structures may have to account for these exceptions, likely using Dynkin’swork on classifying maximal subgroups of semisimple Lie groups in [Dyn00a,Dyn00b].

Non-geometries as base spaces of fiber bundles. Even when invariant fiber bundle structuresexist, a number of compliactions prevent the classification from having a straightforward recursivesolution. Filipkiewicz warns in [Fil83, Prop. 2.1.3] that the base space of an invariant fiber bundlestructure may fail to be a geometry due to noncompact point stabilizers—e.g. the action of PSL(2,C)on S2 ∼= CP1 makes T 1H3 = PSL(2,C)/PSO(2) a fiber bundle over S2. Even when point stabilizersare compact, the base may fail to be maximal (e.g. Heis5 fibers over E4 with U(2) point stabilizers)or a model geometry (e.g. Sol3 over Aff+R, which cannot admit a lattice since it is not unimodular).

2.3 Examples that clarify how the classification is organized

The eight categories of Theorem 1.2 and the grouping of geometries into parametrized familiesinvolved some arbitrary choices. This section discusses the chosen method of organization and somevariations.

The omission of some spaces that one might have guessed. Some of the categories inThm. 1.2 are conspicuously missing geometries that happen to be non-model or non-maximal.

3. The tautological unit circle bundle U(3)/U(2) over CP2 is non-maximal, being equivariantlydiffeomorphic to S5. The two other unit tangent bundles of 3-dimensional spaces of constantcurvature are also non-maximal.

T 1(S3) = SO(4)/SO(2) ∼= S2 × S3 T 1(E3) = R3 o SO(3)/SO(2) ∼= S2 × E3

7. Many of the solvable Lie groups arising from the list in [PSWZ76, Table II] are not unimodularand hence do not admit lattices.

8. Every product geometry with multiple Euclidean factors is non-maximal—but all other prod-ucts are maximal, usually as a consequence of the de Rham decomposition theorem. (See[Gen16b, Prop. 3.12] in Part III.)

6

Counting the geometries and families. The list given in Thm. 1.2 includes 53 individualgeometries and the following 6 infinite families of geometries.

L(a; 1)×S1 L(b; 1), a ≤ b coprime positive integers

S̃L2 ×α S3, 0 < α <∞

S̃L2 ×α S̃L2, 0 < α ≤ 1

R4 oRetA

, eA semisimple integer matrix with 4 real eigenvalues

R4 oRetA

, eA semisimple integer matrix with 2 real eigenvalues

Sol4m,n × E, m, n ∈ Z

To some extent, this count depends on interpretation. The first three families could be expandedto include products of spheres and hyperbolic spaces, while the last three could be unified with

R4 oRx−1, x−1, x+1, x+1

to form one large family with a name like Sol5m,n,p (where m, n, and p are the

middle coefficients of the characteristic polynomial of eA). Indeed, [PSWZ76, Table II] suggeststhis latter unification by listing all semidirect products R4oR with diagonalizable action under thefamily A5,7. We keep the subfamilies of Sol5m,n,p separate since their point stabilizers have differentdimensions.

3 Overview of method

The classification of 5-dimensional geometries M = G/Gp begins, following Thurston [Thu97, §3.8]and [Fil83, §1.2], by using the representation theory of compact groups to list the subgroups Gp ⊆SO(TpM) that could be point stabilizers (Figure 3.1).

Figure 3.1: Closed connected subgroups of SO(5), with inclusions. SO(3)5 denotes SO(3) actingon its 5-dimensional irreducible representation; and S1

m/n acts as on the direct sum Vm ⊕ Vn ⊕ Rwhere S1 acts irreducibly on Vm with kernel of order m. See Part II, [Gen16a, Prop. 3.1] for theproof.

SO(5)

SO(4) SO(3)× SO(2) SO(3)5

U(2)

SU(2)

SO(3)

SO(2)× SO(2)

S11

S1m/n S1

0 = SO(2) S11/2

{1}

The problem divides into cases by the action ofGp on the tangent space TpM (the “linear isotropyrepresentation”)—more specifically, by the highest dimension of an irreducible subrepresentation V .

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Table 3.2: Using the classification, non-product geometries can be listed by point stabilizer.

Stabilizer Geometries

SO(5) E5, S5, H5

U(2) Heis5 and ˜U(2, 1)/U(2)

SO(3)5 SL(3,R)/ SO(3) and SU(3)/ SO(3)

SO(2)× SO(2) R4 oRx−1, x−1, x+1, x+1

and the associated bundles (Thm. 1.2(4))

SO(2) R4 oR2 real roots

and R4 oR(x−1)2, x+1, x+1

S11/2 All line bundles over F4 (Thm. 1.2(5))

S11 The two unit tangent bundles (Thm. 1.2(3)),

R4 oRx2, x2

, and (R×Heis3) oRx3→x2→y

{1} The remaining solvable Lie groups

Figure 3.3: Flowchart of the classification. Let V be an irrep in Gp y TpM of maximal dimension.

dimV = ? Solvable Lie groups.

Base space hasinvariant confor-mal structure.

Products of constantcurvature spaces.

Line bundles de-termined by baseand curvature.

Isotropy irre-ducible spaces.

Identificationkey in Fig. 3.4.

Go to Figure 3.6.

First eight geometriesin Thm. 1.2(8)(b).

Listed in Table 3.5.

Listed inThm. 1.2(1–2).

1

2

3

4

5

At the extremes, one can appeal to existing classifications—the classification of strongly isotropyirreducible homogeneous spaces by Manturov [Man61a,Man61b,Man66,Man98], Wolf [Wol68,Wol84],and Krämer [Krä75] when Gp y TpM is irreducible (dimV = 5); and the classification of low-dimensional solvable real Lie algebras by Mubarakzyanov [Mub63] and Dozias [Doz63] if Gp y TpMis trivial (dimV = 1). These cases are handled in Part II [Gen16a], including the production of an

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identification key for trivial-isotropy geometries (Fig. 3.4).

Figure 3.4: Identification key for solvable geometries G = G/{1}.

Lie algebra g is solvable

nilpotent

4-D abelian ideal

g4 6= 0

g4 = 0

no 4-D abelian ideal

g4 6= 0

g4 = 0

non-nilpotent

nilradical R3

nilradical R4

2 Jordan blocks

3 Jordan blocks

4 Jordan blocks

nilradical R⊕ n3

1-D center

2-D center

R3 o {xyz = 1}0

R4 oRx4

R4 oRx3, x

∼= Nil4 × E

Nil4 oR4→3→1

Nil4 oR3→1

R4 oR(x−1)2, (x+1)2

R4 oRx2, x−1, x+1

R4 oRx−a, x−b, x−c, x+a+b+c

(R×Heis3) oRLorentz, y→x1

Sol41 × E

Otherwise Gp y TpM is nontrivial and reducible (2 ≤ dimV ≤ 4). We classify these—the“fibering geometries”—in Part III, starting by proving the existence of a G-invariant fiber bundlestructure on M [Gen16b, Prop. 3.3]. The Uniformization Theorem and version of a theorem byObata and Lelong-Ferrand [Oba73, Lemma 1] imply the base space has an invariant conformalstructure. Beyond this common behavior, the properties of the fibering and the relevant tools varywith the dimension of the subrepresentation V , naturally suggesting the cases in Figure 3.3.

When dimV = 4, the geometries are determined by curvature and base, in a fashion closelyresembling Thurston’s treatment of dimV = 2 and dimM = 3 in [Thu97, Thm. 3.8.4(b)]; Table 3.5lists the results.

Otherwise, we work systematically with G-invariant fiber bundle structures by recasting theproblem as an extension problem for the Lie algebra of G and solving it with the help of Liealgebra cohomology. Over 3-dimensional base spaces there happen to be only products; but over2-dimensional base spaces a daunting array of possibilities requires some attempt to organize theproblem, summarized in Figure 3.6.

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Table 3.5: Geometries with irreducible 4-dimensional isotropy summand

Base Flat (product) Curved

S4 S4 × EE4 non-maximal E5

H4 H4 × ECP2 CP2 × E non-maximal S5

C2 non-maximal E5 Heis5

CH2 CH2 × E ˜U(2, 1)/U(2)

Figure 3.6: Classification strategy for geometries M = G/Gp fibering over 2-D spaces B.

B hasinvariantmetric?

Extensionof Isom0 E2

by R3?

Levi actionnontrivial?

Products and as-sociated bundles.

T 1H3 and somenon-nilpotent solv-able Lie groups.

Some nilpotent Lie groups.

T 1E1,2 and the linebundles over F4.

no

yes

yes

yes—G̃ is an extension of Isom0B

no—then G̃ is a split extension

no—G̃ is a direct product

4 Related work

Classification of compact homogeneous spaces. Gorbatsevich has produced classificationresults for compact homogeneous spaces M by using a fiber bundle described in [GOV93, §II.5.3.2]whose fibers have compact transformation group, whose base is aspherical, and whose total spaceis a finite cover of M . The classification is complete in dimension up to 5 in general, in dimension6 up to finite covers, and in dimension 7 in the aspherical case [Gor12].

Another approach would be to group the problem by the number of isotropy summands. TheRiemannian homogeneous spaces with irreducible isotropy were classified by Manturov [Man61a,Man61b,Man66,Man98], Wolf [Wol68,Wol84], and Krämer [Krä75]; and the compact Riemannian

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homogeneous spaces with two isotropy summands were classified by Dickinson and Kerr in [DK08].

Classification of naturally reductive spaces. The naturally reductive Riemannian homoge-neous spaces G/Gp—those whose geodesics through p are orbits of 1-parameter subgroups tangentto the representation complementary to T1Gp in Gp y T1G (see e.g. [KN69, §X.3])—have beenclassified in dimension 6 by Agricola, Ferreira, and Friedrich [AFF15]; and in lower dimensions bywork of Kowalski and Vanhecke (see [KPV96, §6] for a summary).

The case of dimension 5, in [KV85, Thm. 2.1],1 shares the following features with the classifica-tion of geometries.

1. Everything with SU(2) isotropy is realized by a homogeneous space with U(2) isotropy [KV91,main result (b)].

2. The associated bundle geometries appear as indecomposable naturally reductive spaces.

The differences between geometries and naturally reductive spaces bear mentioning as well:

1. Naturally reductive spaces need not be maximal as geometries, as demonstrated by non-maximal realizations of S3 ∼= S3 o S1/S1 and S5 ∼= SU(3)/ SU(2).

2. Some geometries—particularly those with trivial isotropy—may not be realizable by naturallyreductive spaces. In 3 dimensions, there is just Sol3; and in 4 dimensions, there are F4 and thefour solvable Lie group geometries other than Heis3 × E and E4. In 5 dimensions, there arethe unit tangent bundles T 1H3 and T 1E1,2, the line bundles over F4, the products involvingSol3, and any solvable Lie group geometries that are not E5, Heis5, or a product involvingHeis3.

A chart in [KPV96, 5.1] summarizes the relations between several other classes of spaces.

Other geometric structures. One can replace the assumption of an invariant Riemannian met-ric (compact point stabilizers) with other structures. A number of difficulties may result from this:geodesic completeness may no longer coincide with other notions of completeness (e.g. in [DZ10,Thm. 2.1]); isotropy representations may fail to be semisimple or faithful; and point stabilizers mayfail to act faithfully on tangent spaces, necessitating techniques like those of [vM13]. In spite ofthese challenges, some results are known, such as:

• Using conformal structures without relaxing other assumptions yields only Sn and En: amanifold whose conformal automorphism group acts transitively with the identity componentpreserving no Riemannian metric is conformally equivalent to one of the two, by theorems ofObata [Oba73, Lemma 1] and Lafontaine [Laf88, Thm. D.1].

• The interaction of complex structures and 4-dimensional geometries was investigated by Wallin [Wal86]; and almost-complex structures on homogeneous spaces up to dimension 6 areclassified by Alekseevsky, Kruglikov, and Winther in [AKW14] with the additional assumptionthat point stabilizers are semisimple.

1 The changes in the corrected version [KPV96, 6.4] appear to amount to (1) changing “symmetric” and “de-composable” to “locally symmetric” and “locally decomposable” and (2) changing the rational parameter to a realparameter in the Type II family (Heisenberg-group-like bundles).

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• Loosely analogous to almost-complex structures are 5-manifolds whose structure group can bereduced to SO(3)5. The classification of such structures in the integrable case by Bobieńskiand Nurowski [BN07, Thm. 4.7] offers an alternative path to the classification of isotropy-irreducible geometries, and there are some further classification results in the non-integrablecase by Chiossi and Fino in [FC07] and by Agricola, Becker-Bender, and Friedrich in [ABBF11].

• The pseudo-Riemannian geometries were classified in dimension 3 by Dumitrescu and Zeghibin [DZ10]; and the pseudo-Riemannian naturally reductive spaces were classified in dimension4 by Batat, López, and María [BLM15].

5 References

[ABBF11] I. Agricola, J. Becker-Bender, and T. Friedrich, On the topology and the geometry ofso(3)-manifolds, Annals of Global Analysis and Geometry 40 (2011), 67–84, arXiv:1010.0260.

[AFF15] I. Agricola, A. C. Ferreira, and T. Friedrich, The classification of naturally reductivehomogeneous spaces in dimensions n ≤ 6, Differential Geometry and its Applications 39(2015), 59–92.

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